Question

Simplify the expression (-6-3i)(5+6i)

Answer 1

-12 - 51i

Explanation:

`original:(-6-3i)*(5+6i)\\FOIL:(-6)(5)+(-6)(6i)+(-3)(5)+(-3i)(6i)\\Multiply:(-30)+(-36i)+(-15i)+(18i^2)\\Combine\ Like\ Terms:(-30) + (-51i)+(-18i^2)\\i^2\ =\ -1:(-30)+(-51i)+(-18)(-1)\\Multiply:(-30)+(-51i)+(18)\\Combine\ Like\ Terms:(-12) + (-51i)\\Simplify:-12-51i`

Thing to remember:

`i = √(-1) \\i^2 = √(-1)*√(-1) = √(-1)^2 = -1\\i^3 = √(-1)*√(-1)*√(-1) = i^2 * i = (-1)*i = -i\\i^4 = √(-1)*√(-1)*√(-1)*√(-1) = √(-1)^2*√(-1)^2 = (-1)*(-1) = 1\\`

And all powers over 4 repeat such that
`i^x\ such\ that\ x\mod4 = y` is the same as
`i^y\\`:

i.e.:

`i^6 = i^(4+2)= i^4*i^2 = 1*-1 = -1\\\\i^9 = i^(4+4+1)=i^4*i^4*i^1=(1)(1)(i) = i`

NOTE: x mod 4 simply means the remainder when x is divided by 4.

i.e.:

`7\mod4 = 3\\8\mod4 = 0\\13\mod4 = 1`